Skewing

Not to be confused with Skew polytope or Blending (skew polytopes).

Skewing, or skew halving, is an operation that can be performed on a regular polyhedron of Schläfli type {p,4}. It is represented with the symbol σ , and a polyhedron obtained from skewing will be self-Petrie.

It is equivalent to the composed operation π∘δ∘η∘π∘δ, although in some cases the intermediate stages of this process may be degenerate or impossible to realize, even though the end result can be realized in some cases (e.g. the skew muoctahedron).

Definition

Abstract polytopes

The skewing of a regular polyhedron with distinguished generators $\left(\rho _{0},\rho _{1},\rho _{2}\right)$ is the regular polyhedron generated by the distinguished generators:

$\left(\rho _{1},\rho _{0}\rho _{2},(\rho _{1}\rho _{2})^{2}\right)$

Generalized skewing

There also exists a generalized skewing operation $\sigma _{k,m}$ defined on polyhedra with a $2(k+m)$-gonal vertex figure. It can be defined in terms of the generalized halving operation as $\pi \delta \eta _{k,m}\pi \delta$.